In operation, there are usually four moveable chasers that cut the thread then back away from the work to permit rapid removal of the tool.
When sufficient length of thread has been cut the die head will open allowing for rapid retraction of the head without interference with the newly formed thread.
The die head shown cuts an outside thread.
There are also collapsible die heads that are used to cut an internal thread.
The bottom picture shows a Die Head used for Thread Rolling.